{{{id=1| x, a0, a1, a2 = var('x, a0, a1, a2') solve([a2*x^2 + a1*x + a0 == 0],x) /// $\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = -\frac{a_{1} + \sqrt{-4 \, a_{0} a_{2} + a_{1}^{2}}}{2 \, a_{2}}, x = -\frac{a_{1} - \sqrt{-4 \, a_{0} a_{2} + a_{1}^{2}}}{2 \, a_{2}}\right]$ }}} {{{id=2| x, a0, a1, a2, a3 = var('x, a0, a1, a2, a3') solve([a3*x^3+a2*x^2+a1*x^1+a0 ==0],x) /// $\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = -\frac{1}{2} \, {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{\sqrt{9 \, a_{0}^{2} a_{3}^{2} + \frac{4}{3} \, a_{0} a_{2}^{3} - \frac{1}{3} \, a_{1}^{2} a_{2}^{2} - \frac{2}{3} \, {\left(9 \, a_{0} a_{1} a_{2} - 2 \, a_{1}^{3}\right)} a_{3}}}{6 \, a_{3}^{2}} - \frac{27 \, a_{0} a_{3}^{2} - 9 \, a_{1} a_{2} a_{3} + 2 \, a_{2}^{3}}{54 \, a_{3}^{3}}\right)}^{\left(\frac{1}{3}\right)} - \frac{a_{2}}{3 \, a_{3}} + \frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(3 \, a_{1} a_{3} - a_{2}^{2}\right)}}{18 \, {\left(\frac{\sqrt{9 \, a_{0}^{2} a_{3}^{2} + \frac{4}{3} \, a_{0} a_{2}^{3} - \frac{1}{3} \, a_{1}^{2} a_{2}^{2} - \frac{2}{3} \, {\left(9 \, a_{0} a_{1} a_{2} - 2 \, a_{1}^{3}\right)} a_{3}}}{6 \, a_{3}^{2}} - \frac{27 \, a_{0} a_{3}^{2} - 9 \, a_{1} a_{2} a_{3} + 2 \, a_{2}^{3}}{54 \, a_{3}^{3}}\right)}^{\left(\frac{1}{3}\right)} a_{3}^{2}}, x = -\frac{1}{2} \, {\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{\sqrt{9 \, a_{0}^{2} a_{3}^{2} + \frac{4}{3} \, a_{0} a_{2}^{3} - \frac{1}{3} \, a_{1}^{2} a_{2}^{2} - \frac{2}{3} \, {\left(9 \, a_{0} a_{1} a_{2} - 2 \, a_{1}^{3}\right)} a_{3}}}{6 \, a_{3}^{2}} - \frac{27 \, a_{0} a_{3}^{2} - 9 \, a_{1} a_{2} a_{3} + 2 \, a_{2}^{3}}{54 \, a_{3}^{3}}\right)}^{\left(\frac{1}{3}\right)} - \frac{a_{2}}{3 \, a_{3}} + \frac{{\left(i \, \sqrt{3} + 1\right)} {\left(3 \, a_{1} a_{3} - a_{2}^{2}\right)}}{18 \, {\left(\frac{\sqrt{9 \, a_{0}^{2} a_{3}^{2} + \frac{4}{3} \, a_{0} a_{2}^{3} - \frac{1}{3} \, a_{1}^{2} a_{2}^{2} - \frac{2}{3} \, {\left(9 \, a_{0} a_{1} a_{2} - 2 \, a_{1}^{3}\right)} a_{3}}}{6 \, a_{3}^{2}} - \frac{27 \, a_{0} a_{3}^{2} - 9 \, a_{1} a_{2} a_{3} + 2 \, a_{2}^{3}}{54 \, a_{3}^{3}}\right)}^{\left(\frac{1}{3}\right)} a_{3}^{2}}, x = {\left(\frac{\sqrt{9 \, a_{0}^{2} a_{3}^{2} + \frac{4}{3} \, a_{0} a_{2}^{3} - \frac{1}{3} \, a_{1}^{2} a_{2}^{2} - \frac{2}{3} \, {\left(9 \, a_{0} a_{1} a_{2} - 2 \, a_{1}^{3}\right)} a_{3}}}{6 \, a_{3}^{2}} - \frac{27 \, a_{0} a_{3}^{2} - 9 \, a_{1} a_{2} a_{3} + 2 \, a_{2}^{3}}{54 \, a_{3}^{3}}\right)}^{\left(\frac{1}{3}\right)} - \frac{a_{2}}{3 \, a_{3}} - \frac{3 \, a_{1} a_{3} - a_{2}^{2}}{9 \, {\left(\frac{\sqrt{9 \, a_{0}^{2} a_{3}^{2} + \frac{4}{3} \, a_{0} a_{2}^{3} - \frac{1}{3} \, a_{1}^{2} a_{2}^{2} - \frac{2}{3} \, {\left(9 \, a_{0} a_{1} a_{2} - 2 \, a_{1}^{3}\right)} a_{3}}}{6 \, a_{3}^{2}} - \frac{27 \, a_{0} a_{3}^{2} - 9 \, a_{1} a_{2} a_{3} + 2 \, a_{2}^{3}}{54 \, a_{3}^{3}}\right)}^{\left(\frac{1}{3}\right)} a_{3}^{2}}\right]$ }}} {{{id=4| f(x)=(x^2+x-3)/(x+3) f.partial_fraction() /// $\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x + \frac{3}{x + 3} - 2$ }}} {{{id=5| S = RR['x'] f=x^5+4*x^4-3*x^2-2*x+1 g=x^3+3*x^2-4 def poldiv(f,g): if f.degree(x) < g.degree(x): return f else: m=f.degree(x) n=g.degree(x) return f-g*f.coefficient(x^m)/g.coefficient(x^n)*x^(m-n) /// }}} {{{id=10| expand(poldiv(f,g)) /// $\newcommand{\Bold}[1]{\mathbf{#1}}x^{4} + x^{2} - 2 \, x + 1$ }}} {{{id=8| K=[(f,0)] m=f.degree(x) n=g.degree(x) h=f while m >= n: K.append((expand(poldiv(h,g)),expand(h.coefficient(x^m)/g.coefficient(x^n)*x^(m-n)))) h=expand(poldiv(h,g)) print h m=m-1 /// x^4 + x^2 - 2*x + 1 -3*x^3 + x^2 + 2*x + 1 10*x^2 + 2*x - 11 }}} {{{id=9| K /// $\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(x^{5} + 4 \, x^{4} - 3 \, x^{2} - 2 \, x + 1, 0\right), \left(x^{4} + x^{2} - 2 \, x + 1, x^{2}\right), \left(-3 \, x^{3} + x^{2} + 2 \, x + 1, x\right), \left(10 \, x^{2} + 2 \, x - 11, -3\right)\right]$ }}} {{{id=16| h=f/g h.partial_fraction() /// $\newcommand{\Bold}[1]{\mathbf{#1}}x + \frac{1}{9 \, {\left(x - 1\right)}} + \frac{89}{9 \, {\left(x + 2\right)}} - \frac{25}{3 \, {\left(x + 2\right)}^{2}} + x^{2} - 3$ }}} {{{id=17| /// }}}